Spectral geometry of saturated Fell bundles

Let $\mathcal{B}=(B_g{)}_{g\in G}$ be a saturated Fell bundle over a countable discrete group, equipped with a spectral triple. We give explicit sufficient conditions for lifting this triple to the reduced cross-sectional C\Star-algebra $C_r^{\ast }(\mathcal{B})$. Using finite projective realizations of the homogeneous Morita equivalence $B_e$-bimodules, we transport the base Dirac operator to the individual fibres and assemble the resulting operators into a fibrewise Dirac operator. We introduce a spectral-regularity condition ensuring bounded commutators and give a sufficient criterion in terms of uniform derivative bounds on smooth factor-system data. Combining the fibrewise operator with a length operator associated with a proper translation-bounded matrix-valued length function yields the lifted spectral triple. When $B_e$ is separable, the lifted triple, with a suitable grading, represents the Kasparov product of the vertical class with the class of the unit-fibre triple. We also establish a summability criterion and sufficient conditions for stability under central cocycle twisting. A range of examples illustrates the scope of the construction, including twisted crossed products, free compact Abelian actions, and Fell bundles with nonfree homogeneous fibres.